Quadrature Amplitude Modulation (QAM) can be used to represent data by changing, or modulating, the amplitude of two carrier waves, which are out of phase with each other by 90 degrees and are thus called quadrature carriers. The quadrature carriers can be modulated in amplitude to represent digital symbols being transmitted. When the amplitude of modulation in the two quadrature carriers for a symbol is represented along the real and imaginary axes in a complex plane, the symbol can be represented as a point in the complex plane. A set of symbols used in a QAM scheme can be collective called a constellation. A constellation diagram shows the set of symbols in the complex plane.
A rectangular QAM constellation includes a set of symbols arranged on a rectangular grid. Rectangular QAM constellations may not be optimal in that the points in the constellation do not maximally space from each other. None rectangular QAM constellations may also be used to improve separation, but they are harder to modulate and demodulate than rectangular QAM constellations.
As the bandwidth demand increases, higher and higher QAM sizes have been adopted in many latest applications, such as DVB-C receiver and cable modem. The specifications for such applications include the ability to handle severe channel conditions such as low signal to noise ratio (SNR) or large echoes. To mitigate such impairments on the channel and to correctly recover the transmitted signal, various channel equalization technologies are employed.
Signals transmitted through a transmission channel suffer from non-ideal channel characteristics such as Additive White Gaussian Noise (AWGN), Inter Symbol Interference (ISI), fading, and phase distortion, etc. The transmitted signals can be distorted by the channel characteristics, which is typically unknown.
Equalization is a technique used to reduce distortion and compensate for signal loss (attenuation). Typically, an equalizer uses an adjustable filter which is adjusted to compensate the unknown channel characteristics. Blind equalization is a type of technology, which does not use any training sequence and thus reduces the system overhead. Blind equalization has been widely used to adapt the receiver to the channel conditions. Many blind equalization algorithms have been developed.
C. R. Johnson, et al., described a Constant Modulus Algorithm (CMA) in “Blind Equalization using the constant modulus criterion: a review, Proc. of IEEE, Vol. 86, Oct. 1998.
Constant Modulus Algorithm (CMA) is a simple and effective way to achieve channel equalization. A Constant Modulus Algorithm minimizes an error function for equalization. The error function is based on the difference between the equalizer output and a constant constellation radius:[|y|p−K]q 
where y is the equalizer output; K a constant; and p and q are typically integers.
FIG. 1 shows a block diagram of a conventional CMA-based blind equalizer. In FIG. 1, the adjustable filter (101) has a number of coefficients, which may also be referred as tap weights, which can be used to adjust the transfer function of the equalizer. The input signal to the adjustable filter (101) may be distorted due to the unknown channel characteristics. The adaptation engine (109) adjusts the tap weights according to the error generator (105) to reduce the error between the output of the adjustable filter (101) and the constant modulus (107). The decision engine (103) identifies the symbol being transmitted from the output of the adjustable filter (101) to generate the decision output (103). The tap weights are continuously adjusted by the adaptation engine (109) to reduce the error until the equalizer converges.
The conventional CMA-based blind equalizer has a drawback of low convergence rate. When the channel condition becomes severer, the convergence becomes more difficult. To enhance the convergence speed, the conventional CMA was modified to develop improved algorithms, such as a Sato algorithm (see, e.g., M. Goursat, et al., in “Blind Equalizers, IEEE Trans. of Communications, Vol. COM-28, August 1984) and a “stop-and-go” decision-directed algorithm (see, e.g., G. Picchi, et al, in “Blind equalization and carrier recovery using a ‘stop-and-go’ decision-directed algorithm, IEEE Trans. Of Communications, Vol. COM-35, September 1987).
M. J. Ready and R. P. Gooch describes a multi-modulus algorithm in “blind equalization based on radius directed adaptation, Proc. 1990 IEEE Int. Conf. Acoust., Speech, Signal Processing, Albuquerque, NM, PP 1699-1702, 1990, in which radius directed adaptation is based on the known modulus of the constellation symbol radii. The error function is based on the difference between the equalizer output and the nearest constellation radius:[|y|p−Kd]q 
where y is the equalizer output; Kd is the radii of the nearest constellation symbol for the equalizer output y; and common values for (p, q) are (1, 1), (1, 2), (2, 1), (2, 2), etc.